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Real Analysis Mathematical Knowledge for Teaching: an Investigation

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Real Analysis Mathematical Knowledge for Teaching: an Investigation IUMPST: The Journal. Vol 1 (Content Knowledge), February 2021 [www.k-12prep.math.ttu.edu] ISSN 2165-7874 REAL ANALYSIS MATHEMATICAL KNOWLEDGE FOR TEACHING: AN INVESTIGATION Blain Patterson Virginia Military Institute [email protected] The goal of this research was to investigate the relationship between real analysis content and high school mathematics teaching so that we can ultimately better prepare our teachers to teach high school mathematics. Specifically, I investigated the following research questions. (1) What connections between real analysis and high school mathematics content do teachers make when solving tasks? (2) What real analysis content is potentially used by mathematics teachers during the instructional process? Keywords: Teacher Content Knowledge, Real Analysis, Mathematical Understanding for Secondary Teachers (MUST) Introduction What do mathematics teachers need to know to be successful in the classroom? This question has been at the forefront of mathematics education research for several years. Clearly, high school mathematics teachers should have a deep understanding of the material they teach, such as algebra, geometry, functions, probability, and statistics (Association of Mathematics Teacher Educators, 2017). However, only having knowledge of the content being taught may lead to various pedagogical difficulties, such as primarily focusing on procedural fluency rather than conceptual understanding (Ma, 1999). The general perception by mathematicians and mathematics educators alike is that teachers should have some knowledge of mathematics beyond what they teach (Association of Mathematics Teacher Educators, 2017; Conference Board of the Mathematical Sciences, 2012, Wasserman & Stockton, 2013). The rationale being that concepts from advanced mathematics courses, such as abstract algebra and real analysis, are connected to high school mathematics content (Wasserman, Fukawa-Connelly, Villanueva, Meja-Ramos, & Weber, 2017). For example, the field axioms discussed in abstract algebra surface as the properties of equality in high school mathematics when solving equations (Wasserman, 2016). If explicit connections exist between the content of real analysis and high school mathematics, it is essential that those who teach calculus, precalculus, and algebra have a firm understanding of this subject (Wasserman et al., 2017). However, there exists little research on the connection between learning real analysis and teaching high school mathematics, despite findings from various studies that assert student achievement is related to the content knowledge of their teachers (Ball, Thames, & Phelps, 2008; Begle, 1972; Hill, Rowan, & Ball, 2005; Monk, 1994). These studies focus on either pedagogical content knowledge or the relationship between knowing abstract algebra and teaching high school algebra. Although these studies have made significant contributions to our understanding of the need for strong content knowledge for teachers, we still know very little about the relationship between learning real analysis and teaching high school mathematics. Therefore, the goal of this study was to investigate this relationship so that we can ultimately better prepare teachers to teach high school mathematics. B, Patterson: Real Analysis Mathematical Knowledge for Teaching: An Investigation Literature Review The Mathematical Understanding for Secondary Teachers (MUST) framework developed by Heid, Wilson, and Blume (2015) consists of the combination of the mathematical proficiency, mathematical activity, and mathematical context perspectives of teaching mathematics. Mathematical proficiency includes conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, productive disposition, as well as historical and cultural knowledge. Engaging in mathematical activity can be thought of as doing mathematics, with the emphasis is on those mathematical activities that teachers employ and that they want their students to learn (Heid, Wilson, & Blume, 2015). This includes activities such as mathematical noticing, reasoning, and creating. Unlike mathematical proficiency and activity, which are present in a variety of scientific fields, mathematical context can be described as understanding how students think about mathematics (Heid, Wilson, & Blume, 2015). In addition to being able to know and do mathematics, teachers must also be able to facilitate the development of their student’s mathematical proficiency and activity. For example, this may include probing mathematical ideas, understanding the mathematical thinking of students, knowing and using the curriculum, and assessing the mathematical knowledge of students (Heid & Wilson, 2016). According to Heid and Wilson (2016), the combination of the mathematical proficiency, mathematical activity, and mathematical context perspectives together form a picture of the mathematical knowledge required to teach high school. Moreover, understanding according to the MUST framework is not simply the sum of knowing mathematics and knowing how to teach. The task of teaching mathematics cannot be partitioned into such simple categories; rather, teaching mathematics requires a unique combination of the two. According to the Standards for Preparing Teachers of Mathematics developed by the Association of Mathematics Teacher Educators (2017), all teachers should possess robust knowledge of both mathematical and statistical concepts that form a foundation of what they teach. Additionally, all teachers should have pedagogical knowledge, including effective and equitable teaching practices as well as a firm understanding of how students think about mathematics (Association of Mathematics Teacher Educators, 2017; Conference Board of the Mathematical Sciences, 2012). For high school mathematics teachers, this means they must have a deep understanding of single- and multivariable calculus, probability and statistics, abstract algebra, real analysis, modeling, differential equations, number theory, and the history of mathematics. Although each of these subjects may be of equal importance, the focus of this study will be to investigate teacher knowledge of real analysis in relation to classroom teaching. Real analysis is a course that nearly all mathematics majors and some mathematics education majors are required to take (Conference Board of the Mathematical Sciences, 2012). Standard topics covered in real analysis include the real number system, functions and limits, topology of the real numbers, continuity, differential and integral calculus for functions of one variable, infinite series, and uniform convergence (Bartle & Sherbert, 2011). This course is often viewed by pre-service high school mathematics teachers as daunting and disconnected from practice (Goulding, Hatch, & Rodd, 2003; Wasserman, Villanueva, Mejia-Ramos, & Weber, 2015). However, perceptions of this disconnect are incorrect since there are many explicit connections between what is learned in an introductory real analysis course and what is taught in high school mathematics courses (Wasserman et al., 2017). Students in real analysis study the structure of the real number line and its subsets (Bartle & Sherbert, 2011). Analogously, high school mathematics teachers must develop student conceptions of the real number system as early as algebra. The convergence of sequences, which 2 Issues in the Undergraduate Mathematics Preparation of School Teachers ISSN 2165-7874 is studied rigorously in real analysis, also appears in the high school curriculum. Those who teach precalculus need to have a firm understanding of limits and continuous functions, both of which are studied intensely in an introductory real analysis course. Concepts that play a major role in calculus, such as differentiation, integration, and infinite series, make up the standard curriculum for the typical real analysis course (Bartle & Sherbert, 2011). Since these topics are foundational in the study of calculus, one may argue that calculus teachers ought to have a background in real analysis. However, let us consider the average high school mathematics teacher, who teaches courses such as algebra, geometry, and precalculus (not calculus). How can taking a course in real analysis benefit this teacher? Methods The purpose of this study was to investigate the relationship between knowledge of real analysis and classroom teaching in order to better understand how studying advanced mathematics can help improve and support the development of high-quality secondary mathematics teachers. By doing this, we may be able to determine if it is worthwhile for teachers to take advanced mathematics courses, and if so, what specifically about these courses is useful for teachers. This study serves as an initial step of this goal, by investigating the connections between real analysis and high school mathematics and how these connections can inform classroom teaching. Specifically, the following research questions will be addressed. 1. What connections between real analysis and high school mathematics content do teachers make when solving tasks? How can these connections be characterized in terms of mathematical proficiency and mathematical activity? 2. What real analysis content is potentially used by mathematics teachers during the instructional process? How can the use of this content knowledge be characterized in terms of mathematical proficiency, mathematical activity, and mathematical context? This study took place over the course of the 2018-2019 academic year
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